ИСТИНА

Some of the new parameterization issues for three-dimensional thin body areas are considered, as well as some of the geometrical characteristics of the parameterization are given [1, 2]. The statements of the initial boundary value problems of three-dimensional linear classical and micropolar theories of viscoelastic bodies are formulated, on the basis of which the corresponding statements of the initial boundary value problems of three-dimensional linear classical and micropolar theories of viscoelastic thin bodies are formulated under the new parametrization of the thin body region. From the latter statements, in turn, we obtained the statements of initial-boundary value problems in moments with respect to systems of orthogonal polynomials and, in particular, with respect to a system of Legendre polynomials. The statements of the initial boundary value problems are also considered in the case of classical theories with respect to the displacement vector and in the case of micropolar theories with respect to the displacement and rotation vectors. The constitutive relations bodies are recorded with the help of tensor and tensor-block matrices, and also with the help of canonical representations [2] of these tensor objects. In addition, static boundary conditions and equations of motion and equilibrium are represented by differential tensor operators in the case of classical theory and differential tensor-block matrix operators in the case of micropolar theory. Tensor operators of cofactors for tensor operators of equations of motion and equilibrium and tensor operators of stresses are constructed, which allow solving the problems of decomposition of initial-boundary value problems of the linear classical and micropolar theories of viscoelastic bodies [1, 3]. It should be noted that all of the above is easily applicable to theories of other rheology bodies, including second gradient bodies. Acknowledgements: the work was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement №075-15-2019-1621 References 1.Никабадзе М.У. Развитие метода ортогональных полиномов в механике микрополярных и классических упругих тонких тел. М.: Изд-во Попечительского совета мех.-мат. ф-та МГУ. 2014. 515 с. https://istina.msu.ru/publications/book/6738800/ 2.M. U. Nikabadze Topics on tensor calculus with applications to mechanics. Journal of Mathematical Sciences, Vol. 225, No. 1, 2017. 194 p. DOI: 10.1007/s10958-017-3467-4 3.Nikabadze M., Ulukhanyan A. On the Decomposition of Equations of Elastisity and Thin body Theoty. Lobachevskii Journal of Mathematics. 2020; 41(10), 2059-2074. https://dx.doi.org/10.1134/S1995080220100145/